math 10Q fast help (partial diff questions)
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sampleee.pdf
AME 500B Final Exam (3 Hours)
5/10/20
Before beginning any problem, read the entire exam. Do the problem that seems simplest first. Please begin each problem on a separate page of your own paper. Open book and notes, but no internet. Submit to Final Submission Box
(20) 1. Consider the following wave equation for an infinite string:
( ) 2 2
2 2 2 , 0c u x tt x
∂ ∂ − = ∂ ∂
.
Using the coordinate transformation ,x ct x ctξ η≡ + ≡ −
show that ( ) ( )( )2 , , ,
0 u x tξ η ξ η
ξ η ∂
= ∂ ∂
.
(15) 2. If ( ) ( ), 0u x f x= and ( ) ( )
0
,
t
u x t g x
t =
∂ =
∂ , show the solution to the wave
equation of Problem 1 is
( ) ( ) ( ) ( )1 1, 2 2
x ct
x ct u x t f x ct f x ct dt g t
c
+
−
′ ′= + + − + ∫ .
(20) 3. Solve the wave equation
( ) 2 2
2 2 2 , 0c u x tt x
∂ ∂ − = ∂ ∂
subject to the boundary conditions ( ) ( )0, , 0u t u L t= =
and initial conditions
( ) ( ) ( ) ( ) 0
, 0 , 0 ,
t
u x u x f x g x
t =
∂ = =
∂
to show that the solution satisfies ( ) ( ) ( ), vu x t x ct w x ct= + + − .
(10) 4. Is the polynomial 2 2( , ) 2P x y x y ixy= + −
analytic? What two changes will make this polynomial analytic? (15) 5. If f(z) is an analytic function, show that
( ) ( ) ( ) 22
2 f z f z f z
x y
∂ ∂ ′+ = ∂ ∂ .
Hint: ( ) vuf z i x x ∂ ∂
′ = + ∂ ∂
(10) 6. Criticize the following argument: Since
1
0
1
1 ; 1
k
k
k
k
z z
z z
z z
∞
=
∞ −
=
= −
= + −
∑
∑
therefore
0 1 1
z z z z + =
− − .
(20) 7. Evaluate the following integral for k < 1:
( ) 2
0
1 1 cos
I d k
π
θ θ
= +
∫ .
(20) 8. Consider the following Sturm-Liouville problem:
( ) ( )2 0, 0 1 d xd
x k x dx dx
β φ φ β
+ = < <
.
satisfying homogeneous BC. State at least 7 features of the solution that are known without having to solve the equation.
(25) 9.a. Directly from the solution of the following 2-D heat transfer equation:
( ) 2 2
2 2 , , 0, 0 , 0u x y t x a y bt x y ∂ ∂ ∂
− − = < < < < ∂ ∂ ∂
with homogeneous BC ( ) ( ) ( ) ( ) 0, , , , 0
, 0, , , 0
u y t u a y t
u x t u x b t
= =
= =
and IC ( ) ( ) ( ), , 0u x y f x g y= ,
show that ( ) ( ) ( )1 2, , , ,u x y t u x t u y t= ,
where
( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
2
12
1 1
1
2
22
2 2
2
, 0, 0 ;
0, 0, , 0
, 0
, 0, 0
0, 0, , 0
, 0 .
u x t x a t x
u t u a t
u x f x
u y t y b t y
u t u b t
u y g y
∂ ∂ − = < < ∂ ∂
= =
=
∂ ∂ − = < < ∂ ∂
= =
=
b. Predict what the solution will be for 3D with the same BC in the third
dimension and a corresponding product IC.
